No pics to look at.
If the connector is an RCA plug and you broke the center pin it can be extracted with a bit of care. A broken input doesn't affect the audio output jacks. Just makes that input unusable.
Without the make and model of the projector and stereo we can't know what connections you have can we?
In any case you don't need to get your audio from the projector. You need to get it from your source or sources. If you have one source with the right kind of audio output for your stereo then use it. If you have multiple sources you can get an AV input selector or HDMI input selector with audio extraction. The extractor usually has digital and analog audio output. The cables might be easier this way too.